A sphere has two surfaces. A sheet of paper has two surfaces. The first one sided surface was discovered by A.F Mobius and bears his name Mobius strip.

A model can be obtained by taking a paper strip ,giving it a half twist(180 degree), and then attaching the two ends. If you start drawing a line down its center along a line parallel to its edge of the strip, you'll end up right back where you started. You will never cross an edge. To compare this you can make a cylinder by attaching the ends of the paper without any twist and draw a line like you did before. See the difference.In Euclidean space there are two types of mobius strips depending upon the direction(clockwise or anticlockwise) of the twist.


This illustration shows interlocked gears along the length of a mobius strip.

The mobius strip has several properties. Try cutting the strip along the line you drew. Instead of getting two separate strips you get one strip with two half twists. Then draw a center line around the resulting band and cut along it. You will get two strips wound around each other.